Simple harmonic motion is physical motion whose acceleration is proportional to displacement from equilibrium and points back toward equilibrium:

$$a=-\omega^2x.$$

This restoring relation produces the harmonic oscillator equation, so the motion is sinusoidal with angular frequency $\omega$.

Mass on an ideal spring

For a mass $m$ attached to a spring of stiffness $k$, Hooke's law gives

$$F=-kx.$$

Combining this with Newton's second law,

$$m\ddot x=-kx,$$

so

$$\omega=\sqrt{\frac{k}{m}}.$$

Increasing stiffness raises the natural frequency; increasing mass lowers it.

Motion through equilibrium

Velocity is largest in magnitude as the oscillator passes through equilibrium. At the turning points $x=\pm A$, velocity is zero and the restoring acceleration has maximum magnitude.

Simple harmonic motion is the physical model produced by an ideal linear restoring force; the sinusoidal solution itself belongs to the underlying harmonic oscillator equation.